Open subgroups of T : Recall that p is a prime number, and T := Gal (L_{∞}/Q_{p}) is the Galois group of L_{∞}, the maximal tamely ramified extension of Q_{p}. Moreover, we have been studying the exact sequence

1 → ∏_{ℓ≠p} Z_{ℓ}(1) → T → Ẑ → 1,

where here we have done the confusing thing of writing the group law of the two flanking groups in the above exact sequence “additively”, and yet we will continue, of course, to think of the group law in the noncomuutative group T as being written “multiplicatively”. The action of the element 1 in Ẑ on ∏_{ℓ≠p} Z_{ℓ}(1) (via first lifting this element to an element ϕ ∈ T, and then considering the conjugation action x ↦ ϕxϕ^{−1} is multiplication by p). The above exact sequence splits (noncanonically), and in fact, any element ϕ ∈ T which lifts 1 ∈ Ẑ effects such a splitting. Any such element ϕ ∈ T will be called “a Frobenius element”. Fix such a “Frobenius element” ϕ, and recall that T may then be described as the profinite group on two generators {τ, ϕ} satisfying only the one relation

(∗) ϕτϕ^{−1} = τ^{p}.

Clearly any element of T can be uniquely represented as a product α · β where α is in the closed subgroup of T generated by τ and β is in the closed subgroup of T generated by ϕ. For any pair of positive integers e, f, with e not divisible by p, consider the closed subgroup T_{e,f} ⊂ T generated by ϕ^{f} and τ^{e}. These generators clearly satisfy the relation

ϕ^{f} · τ^{e} · ϕ^{−f} = (τ^{e})^{q},

where q = p^{f}, and T_{e,f} is the profinite group admitting as presentation those two generators and that one relation.

Exercise: Show that the T_{e,f} (for all positive integers e not divisible by p, and positive integers f) comprise all open subgroups of T of finite index, and the open normal subgroups of finite index are precisely those T_{e,f} where e divides p^{f}. In case T_{e,f} is normal, show that the finite group T/T_{e,f} is a semi-direct product of the cyclic group Z/fZ by a cyclic group C_{e} of order e, where, if we write the group law on C_{e} multiplicatively, the action of the generator 1 ∈ Z/fZ on C_{e} is via “raising to the q = p^{f}-th power”.

The representation theory of T and its subgroups. Let, then F be a local field (an “LF”, say, which will play the lowly role of a field of scalars) and V a finite dimensional vector space over F admitting a (continuous) F -linear T -action.

Claim: Given any compact group Γ acting continuously as a group of F -linear automorphisms of V, there exists an O_{F} -latticeΩ ⊂ V which is stabilized by the action of Γ. Any such representation is equivalent (over F) to a representation given by a continuous homomorphism ρ : Γ → GL_{d}(O_{F}), where d = dim_{F} (V).

So, equivalently, V is equipped with a pair of operators τ, ϕ : V → V which preserve an O_{F} -lattice in V, and which satisfy the formula ϕτϕ^{−1} = τ^{p}. Consider the operator τ : V → V. Suppose that v ∈ V is a (nonzero, of course) eigenvector for τ with eigenvalue λ ∈ F. For integers n ≥ 0, put v_{n} := ϕ^{−n}(v) and calculate using the relation (∗) to find that v_{n} is an eigenvector for τ with eigenvalue λ^{p^{n}} ∈ F. Since we have assumed that V is finite-dimensional, τ can only have a finite number of distinct eigenvalues, giving that λ^{p^{n}} = λ^{p^{m}} for some pair of integers n ≠ m, i.e., any eigenvalue λ of τ is a (p − 1)p^{ν}-th root of unity for some ν. It follows that any eigenvalue of τ is, in fact, a (p − 1)-st root of unity, for τ is in the closed subgroup generated by τ^{p^{ν}}. We have shown, therefore, that the operator τ^{p−1} is unipotent (i.e., is equal to the operator I + N, where I is the identity and N is a nilpotent operator on V).

Exercise: What is the analogous statement for the subgroup T_{e,f}?

The representation theory of the Galois groups.

Now we consider G_{Q_{p}} := Gal(Q̄_{p}/Q_{p}), and recall the exact sequence

1 → P → G_{Q_{p}} → T → 1.

If K/Q_{p} is a finite extension field, with G_{K} ⊂ G_{Q_{p}} its Galois group (equivalently: G_{K} = {g ∈ G_{Q_{p}} | g(x) = x for all x ∈ K}) let I_{K} := I ∩ G_{K}, where I ⊂ G_{Q_{p}} is the inertia subgroup. Put P_{K} := G_{K} ∩ P and T_{K} := the image of G_{K} in T, so that T_{K} = T_{e,f} where f is the absolute residue degree of K and e is the absolute tame (i.e., prime to p) ramification index of K. We will refer to the image of I_{K} in T_{K} as the tame inertia subgroup of T_{K}.

We have the exact sequences

1 → P_{K} → G_{K} → T_{K} → 1.

Exercise for people who know Class Field Theory. Consider the infinite-dimensional vector space V_{K} := Hom_{cont}(P_{K}, Q_{p}) of continuous group-homomorphisms of P_{K} to the additive group of p-adic numbers, Q_{p}. The natural action of T_{K} on P_{K} induces a representation of T_{K} on the Q_{p}-vector space V_{K}. Can you describe this infinite-dimensional representation?

Definition. Let F be a local field (an “LF”) of residual characteristic different from p, and V a finite dimensional vector space over F. Then a continuous G_{K}-action on V is called semi-stable if every element of I acts unipotently on V. It is called potentially semi-stable if there is a finite extension field L/K of K such that the restriction of the G_{K}-action on V to G_{L} is semi-stable.

Proposition. (Grothendieck) Every continuous G_{K} representation on a finite-dimensional vector space over a local field F of residual characteristic different from p is potentially semi-stable.

Proof. Let O_{F} be the ring of integers of F, and k_{F} its the residue field . Let ℓ be the characteristic of k_{F} so that by assumption we have ℓ ≠ p. Let us take the view that our representation is given by a continuous homomorphism (of locally compact groups) ρ : G_{K} → GL_{d}(O_{F}) for some positive integer d, which is legitimate, given the “Claim” above. Make a preliminary finite extension K′/K so that the image of the restriction ρ′ : G_{K′} → GL_{d}(O_{F}) is contained in the kernel of the residual homomorphism GL_{d}(O_{F}) → GL_{d}(k_{F}). Changing notation (from K′ to K) we may simply assume, now, that the image of ρ is contained in the kernel of the residual homomorphism, which is a pro-ℓ-group. Since there are no nontrivial continuous homomorphisms of a pro-p-group to a pro-ℓ-group, the pro-p-group P_{K} is contained in the kernel of the homomorphism ρ; i.e., ρ is induced from a representation of T_{K}. But, after our discussion above, any finite-dimensional representation of T_{K} has the property that there is a finite field extension L/K such that the tame inertia subgroup of L in T_{L} acts via unipotent elements. It follows then that the restriction of ρ to G_{L} is semi-stable.

Putting further squeezes on the continuous representations of G_{K} on finite dimensional vector spaces over local fields F, where the residual characteristics of K and F are different.

Let ℓ be the residue characteristic of F, and, as always, p the residue characteristic of K. I’m running out of convenient letters for prime numbers, but let r be a prime number different from p and ℓ, and let T_{ℓ} denote the quotient group of T that you get when you divide by the (normal, closed) subgroup of T given by ∏_{r≠p,ℓ} Z_{ℓ}(1):

∏_{r≠p,ℓ} Z_{ℓ}(1) ⊂ ∏_{r≠p} Z_{r}(1) ⊂ T.

We have that T_{ℓ} is a semi-direct product of the profinite completion of Z with a chosen generator being given by the Frobenius element ϕ ∈ g) by the free ℓ-adic group on one generator, Z_{ℓ}(1). As before, for K/Q_{p} a finite extension, define the analogous quotient group, T_{ℓ,K}, of G_{K}. The argument given in the Proposition above proves, in fact, that any continuous representation ρ of G_{K} on a finite-dimensional vector space over F has the property that there is a finite field extension L/K such that the restriction of ρ to G_{L} factors through the quotient group, T_{ℓ,L} and the image of the inertia group of L acts unipotently.

Galois cohomology. This is a new section, so we let: K = any field, K_{s}/K a separable algebraic closure of K. Put G_{K} := Gal(K_{s}/K) = lim Gal(L/K) where the projective limit is taken over all finite Galois extensions L/K in K_{s}. We view G_{K} as a compact topological group given its profinite (“Krull) topology.

Definition. An abelian group M together with a linear G_{K}-action on it is called a Galois module over K (cf. Serre: “topological” G_{K}-module) if

M = ⋃ M^{H},

where H runs through all open normal subgroups of G_{K}.

One might make the distinction here between Galois modules and étale abelian sheaves over Spec(K) – the categories of these being, in fact, equivalent. Given an étale abelian sheaf F, you get a G_{K} module M_{F} := F(K_{s}) with its natural G_{K}-action.

H^{q}(G_{K}, M) = ind. lim.H^{q}(G_{K}/H, M^{H}),

where H runs through all open normal subgroups of G_{K}. If we think of M as coming from an étale abelian sheaf F we may write these as the étale cohomology groups H^{q}(Spec K, F).

H^{0}(G_{K}, M) = M^{G_{K}} = Hom_{G_{K}} (Z, M),

and, of course, if M = M_{F} we can think of this as F(K).

Example. When G = G_{k} = Ẑ: e.g., k finite. ϕ ∈ G_{k} a topological generator corresponding to 1 ∈ Ẑ. run through the theory– G_{k}-module M is an abelian group M with an automorphism each of whose orbits are finite. We have

0 → M^{G} → M → M → M_{G} → 0.

Gives cohomology. Snake lemma. If M is finite, then Euler characteristic is 0. Cupproduct. With the natural mappings

M^{G} ⊗ N^{G} → (M ⊗ N)^{G},
M^{G} ⊗ N_{G} → (M ⊗ N)_{G}.
H^{1}(G_{k}, Q/Z) = Q/Z.

This gives duality: If M, N are pontriagin dual, i.e., we have a perfect pairing

M ⊗ N → Q/Z,

then we have a perfect pairing.

H^{i}(G, M) ⊗ H^{1−i}(G, N) → H^{1}(G, Q/Z) = Q/Z.